Abstract
The third stage of oil spreading on water, in which surface-tension force promotes spreading against the resisting viscous effect, is investigated using a similarity solution in combination with an integral boundary-layer technique to solve the unidirectional oil-spreading dynamics problem in the last stage of spreading. The thin layer is assumed to be supplied by oil from a bulk boundary. Using a constitutive equation for oil-film surface tension versus oil-film thickness, analytical solutions near the bulk boundary and near the edge are developed. Using the asymptotic solutions to initiate integration, the differential equations for the oil thickness, oil velocity, and boundary-layer profiles are integrated starting from the leading edge and bulk boundary, which after matching provide a complete solution. The results for the spreading-law prefactors are found to differ by about 10 % from published theoretical results using the same constitutive equation. Using an empirical constitutive equation for oil-film surface tension versus distance from the bulk boundary leads to a spreading-law prefactor that is in excellent agreement with the published experimental result and published theoretical work providing and using the same empirical constitutive equation.
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Abbreviations
- A:
-
Constant in (35)
- a:
-
Constant in (33)
- \(\hbox {a}_\mathrm{i}\) :
-
\(\hbox {i}\)th coefficient of the polynomial approximating the water velocity profile in the y-direction
- B:
-
Constant in the asymptotic form for \(\bar{{\upsigma }}\) (as oil thickness becomes very large), Eq. (40)
- C:
-
Variable in Eq. (22)
- E:
-
Fitting parameter in Eq. (47)
- F:
-
Fitting parameter in Eq. (47)
- h:
-
Oil film thickness
- \(\bar{\hbox {h}}\) :
-
Dimensionless h
- \(\hbox {h}_0 \) :
-
Prefactor, Eq. (41)
- \(\ell \) :
-
Spill extent in the direction of spreading, Fig. 1
- \(\hbox {h}^{{*}}\) :
-
Characteristic oil film thickness
- \(\bar{{\ell }}\) :
-
Dimensionless \(\ell \)
- L:
-
Order of magnitude for the transition spill size
- L. E.:
-
Leading edge
- N:
-
\(= (3 \hbox {n}+1)/2\), Eq. (47)
- n:
-
Fitting parameter between 0 and 0.9
- p:
-
Constant in the asymptotic form for \(\bar{{\upsigma }}\) (as oil thickness becomes very large), Eq. (40)
- r:
-
Velocity power-law exponent (asymptotic solution near the bulk boundary), Eq. (45)
- s:
-
Boundary layer thickness power-law exponent (asymptotic solution near the bulk boundary), Eq. (44)
- S:
-
Spreading coefficient
- t:
-
Time
- \(\bar{\hbox {t}}\) :
-
Dimensionless t
- T:
-
Order of magnitude for the transition time
- U:
-
Oil velocity
- \(\bar{\hbox {U}}\) :
-
Dimensionless U
- u:
-
Horizontal component of the water velocity
- \(\bar{\hbox {u}}\) :
-
Dimensionless u
- \(\hbox {V}_\mathrm{f} \) :
-
Half of the oil-film volume per unit width
- v:
-
Vertical component of the water velocity
- \(\bar{\hbox {v}}\) :
-
Dimensionless v
- x:
-
Horizontal coordinate, Fig. 1
- \(\bar{\hbox {x}}\) :
-
Dimensionless x
- y:
-
Vertical coordinate, Fig. 1
- \(\bar{\hbox {y}}\) :
-
Dimensionless y
- z:
-
Variable in (24)
- \(\upalpha \) :
-
Fig. 3
- \(\upbeta \) :
-
Fig. 4
- \(\Lambda \) :
-
Constant in (39)
- \(\upgamma \) :
-
Constant in (35)
- \(\upsigma \) :
-
Oil-film surface tension
- \(\bar{{\upsigma }}\) :
-
Dimensionless \(\upsigma \)
- \({\upsigma }_\mathrm{oa} \) :
-
Oil–air interfacial tension
- \(\upsigma _\mathrm{ow} \) :
-
Oil–water interfacial tension
- \(\upsigma _\mathrm{wa} \) :
-
Water–air interfacial tension
- \(\updelta \) :
-
Boundary-layer thickness divided by \(\sqrt{{\upnu }_\mathrm{w} \;\hbox {T}}\)
- \(\tilde{\updelta }\) :
-
\(\updelta /\sqrt{\bar{\hbox {t}}}\)
- \(\tilde{\updelta }_\mathrm{s} \) :
-
Square of \(\tilde{\updelta }\)
- \(\upeta \) :
-
Similarity-solution variable, variable in (21)
- \(\upeta _\mathrm{m} \) :
-
Maximum value of \(\upeta \) reached at the leading edge
- \(\bar{\upeta }\) :
-
Ratio \(\upeta /\upeta _\mathrm{m} \)
- \({\uprho }_\mathrm{w} \) :
-
Density of water
- \({\upnu }_\mathrm{w} \) :
-
Kinematic viscosity of water
- \(\upmu _\mathrm{w} \) :
-
Viscosity of water
- \(\uptau _\mathrm{ow} \) :
-
Shear stress at the water–oil interface
- i:
-
Polynomial coefficient index
- m:
-
Maximum (leading-edge value)
- oa:
-
Oil–air interface
- ow:
-
Oil–water interface
- wa:
-
Water–air interface
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Chebbi, R. Spreading of oil on water in the surface-tension regime. Environ Fluid Mech 14, 1443–1455 (2014). https://doi.org/10.1007/s10652-014-9346-3
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DOI: https://doi.org/10.1007/s10652-014-9346-3